A cough for 3 weeks

Hi all! My 1.5 year old has had a wet cough for 3 weeks now. She’s also congested and snotty. We went to the doctor yesterday and she thinks it’s allergies and told us to try Zyrtec. So we started Zyrtec yesterday but I’m really thinking maybe we need antibiotics.

My kid was supposed to start at her new school yesterday but I kept her out due to this cough, as the school won’t allow a child with a cough that disrupts nap time. So I’m trying to really kick this cough’s butt so she can get into school. Do you think I should just ask for the antibiotics although Dr thinks it’s allergies?